What Is a "Nonlinear" Exposure in Value at Risk (VaR)?

When building a portfolio of investments, investors and traders seek to minimize the risk and the potential loss. Traditional practices, such as diversification, help reduce the risk of a portfolio.

To actually reduce the risk of a portfolio and to achieve a point at which a trader would be comfortable with a certain loss, the trader first has to understand what the potential loss of their portfolio is and make adjustments. There are a variety of statistical tools that help traders and investors determine the risk of the portfolio, one of the most common being the value at risk (VaR).

Value at Risk (VaR)

Value at risk (VaR) is a statistical risk management technique that determines the amount of financial risk associated with a portfolio. The VaR of a portfolio measures the amount of potential loss within a specified time period with a degree of confidence. For example, consider a portfolio that has a 1% one-day value at risk of $5 million. With 99% confidence, the expected worst daily loss will not exceed $5 million. There is a 1% chance that the portfolio could lose more than $5 million on any given day.

There are generally two types of risk exposures in a portfolio: linear or nonlinear. Nonlinear risk arises from nonlinear derivatives; those whose payoff changes with time and the location of the strike price to the spot price.

Nonlinear Considerations

Nonlinear risk exposure arises in the VaR calculation of a portfolio of nonlinear derivatives. Nonlinear derivatives, such as options, depend on a variety of characteristics, including implied volatility, time to maturity, underlying asset price, and the current interest rate.

It is difficult to collect the historical data on the returns because the option returns would need to be conditioned on all of the characteristics to use the standard VaR approach. Inputting all of the characteristics associated with options into the Black-Scholes model or another option pricing model causes the models to be nonlinear due to the nature of the derivative. Therefore, the payoff curves are nonlinear because the corresponding value is not proportional to the input due to the time and volatility portion of the model, in particular since options are wasting assets.

The nonlinearity of certain derivatives leads to nonlinear risk exposures in the VaR of a portfolio. Nonlinearity can be witnessed in the payoff diagram of a plain vanilla call option. The payoff diagram has a strong positive convex payoff profile before the option's expiration date, with respect to the stock price.

When the call option reaches a point where the option is in the money, it reaches a point where the payoff becomes linear. Conversely, as a call option becomes increasingly out of the money, the rate at which the option loses money decreases until the option premium is zero.

Kurtosis

If a portfolio includes nonlinear derivatives, such as options, the distribution of the portfolio returns will have a positive or negative skew or high or low kurtosis. The skewness measures the asymmetry of a probability distribution around its mean. Kurtosis measures the distribution around the mean; a high kurtosis has fatter tail ends of the distribution, and a low kurtosis has skinny tail ends of the distribution.

Therefore, it is difficult to use the VaR method that assumes the returns are normally distributed. Instead, the VaR calculation of a portfolio containing nonlinear exposures is usually calculated using Monte Carlo VaR simulations of options pricing models to estimate the VaR of the portfolio.

The Bottom Line

Value at Risk (VaR) is a statistical tool that measures the potential loss of a portfolio at a given time with a certain confidence level. A standard VaR approach does not suit nonlinear derivatives, as their returns are not normally distributed. Other VaR approaches, such as the Monte Carlo VaR, are better suited to predict the measure of loss for irregular distributions of returns.